Set \(c=\mbox{cx}\,_R(M)\) , we argue by induction on \(c\) . If \(c=0\)  then \(\mbox{Pd}\,_R(M)<\infty \)  and so \(\mbox{Pd}\,_R(M)=\mbox{G-dim}\,_R(M)=0\) . Therefore \(\mbox{Tr}\,M=0\)  and we have nothing to prove.
As \(\mbox{Tor}\,_i^R(M,N)=0\)  for all \(i>0\) , \(\mbox{Ext}\,^1_R(\mathcal {T}_{i+1}M,N)=0\)  for all \(i>0\)  by Theorem REF .
Since \(\mbox{Ext}\,^i_R(M,R)=0\)  for all \(i>0\)  then \(\underline{\mathcal {T}_iM}\cong \underline{\Omega \mathcal {T}_{i+1}M}\)  for all \(i>0\)  and so
\(\mbox{Ext}\,^i_R(\mathcal {T}_tM,N)=0 \text{ for all } t>1 \text{ and } 1\le i<t.\) 
Suppose that \(c>0\)  and that \(\eta \in \mbox{Ext}\,^*_R(M,M)\)  reduces the complexity of \(M\) . Consider The exact sequence
\(0\rightarrow M\rightarrow K_{\eta }\rightarrow \Omega ^q_R(M)\rightarrow 0,\) 
where \(|\eta |=q+1\)  and \(\mbox{cx}\,_R(K_{\eta })<c\) . Note that \(\mbox{G-dim}\,_R(K_{\eta })=\mbox{G-dim}\,_R(M)=0\) .
The exact sequence (REF ), induces the long exact sequence
\(\cdots \rightarrow \mbox{Tor}\,_i^R(M,N)\rightarrow \mbox{Tor}\,_i^R(K_{\eta },N)\rightarrow \mbox{Tor}\,_{i+q}^R(M,N)\rightarrow \cdots ,\) 
of homology modules. Therefore, \(\mbox{Tor}\,_i^R(K_{\eta },N)=0\)  for all \(i>0\)  and so by induction hypothesis \(\mbox{Ext}\,^i_R(\mbox{Tr}\,K_{\eta },N)=0\)  for all \(i>0\) .
By [1]}, from the exact sequence (REF ), we obtain the following exact sequence
\( 0\rightarrow {(\Omega ^q_R(M))}^*\rightarrow {K_{\eta }}^*\rightarrow M^*\rightarrow \Omega ^q_R(M))\rightarrow K_{\eta })\rightarrow M)\rightarrow 0\) 
where \(\underline{M)}\cong \underline{\mbox{Tr}\,M}\) , \(\underline{K_{\eta })}\cong \underline{\mbox{Tr}\,K_{\eta }}\)  and \(\underline{\Omega ^q_R(M))}\cong \underline{\mathcal {T}_{q+1}M}\) . Since \(\mbox{Ext}\,^1_R(\Omega ^q_R(M),R)=0\) , we get the exact sequence
\(0\rightarrow \Omega ^q_R(M))\rightarrow K_{\eta })\rightarrow M)\rightarrow 0.\) 
The exact sequence (REF ), induces a long exact sequence
\(\cdots \rightarrow \mbox{Ext}\,^i_R(M),N)\rightarrow \mbox{Ext}\,^i_R(K_{\eta }),N)\rightarrow \mbox{Ext}\,^i_R(\Omega ^q_R(M)),N)\rightarrow \cdots \) 
of cohomology modules.
As \(\mbox{Ext}\,^i_R(\mbox{Tr}\,K_{\eta },N)=0\)  for all \(i>0\) , we obtain from the (REF )
\(\mbox{Ext}\,^{i+1}_R(\mbox{Tr}\,M,N)\cong \mbox{Ext}\,^i_R(\mathcal {T}_{q+1}M,N)\)  for all \(i>0\)  and since
\(\underline{\mathcal {T}_iM}\cong \underline{\Omega \mathcal {T}_{i+1}M}\)  for all \(i>0\) ,
\(\mbox{Ext}\,^i_R(\mathcal {T}_{q+1}M,N)\cong \mbox{Ext}\,^{i+1}_R(\mbox{Tr}\,M,N)\cong \mbox{Ext}\,^{i+q+1}_R(\mathcal {T}_{q+1}M,N) \text{ for all } i>0.\) 
Therefore if \(q>0\)  then by (REF ) and (REF )
\(\mbox{Ext}\,^i_R(\mathcal {T}_{q+1}M,N)=0 \text{ for } i\ne j(q+1) \text{ and } j>0,\) 
\(\mbox{Ext}\,^{q+1}_R(\mathcal {T}_{q+1}M,N)\cong \mbox{Ext}\,^{j(q+1)}_R(\mathcal {T}_{q+1}M,N) \text{ for all } j>0.\) 
By [2]}, there exists an exact sequence
\(0\rightarrow \Omega ^{q+1}_R(K_{\eta })\rightarrow K_{\eta ^2}\oplus F\rightarrow K_{\eta }\rightarrow 0,\) 
where \(F\)  is free. As \(\mbox{G-dim}\,_R(K_{\eta })=0\) , by [1]} we obtain the following exact sequence
\(0\rightarrow K_{\eta })\rightarrow K_{\eta ^2} \oplus F)\rightarrow \Omega ^{q+1}_R(K_{\eta }))\rightarrow 0,\) 
where
\(\underline{K_{\eta })}\cong \underline{\mbox{Tr}\,K_{\eta }}\) , \(\underline{K_{\eta ^2}\oplus F)}\cong \underline{\mbox{Tr}\,K_{\eta ^2}}\)  and
\(\underline{\Omega ^{q+1}_R(K_{\eta }))}\cong \underline{\mathcal {T}_{q+2}K_{\eta }}\) .
The exact sequence (REF ), induces a long exact sequence
\(\cdots \rightarrow \mbox{Ext}\,^i_R(\Omega ^{q+1}_R(K_{\eta })),N)\rightarrow \mbox{Ext}\,^i_R(K_{\eta ^2}\oplus F),N)\rightarrow \mbox{Ext}\,^i_R(K_{\eta }),N)\rightarrow \cdots \) 
of cohomology modules. Note that by the proof of [2]}, \(\Omega ^{q+1}_R(K_{\eta })\)  has also reducible complexity. Therefore, by induction hypothesis, \(\mbox{Ext}\,^i_R(\mathcal {T}_{q+2}K_{\eta },N)=0\)  for all \(i>0\)  and so by (REF ), \(\mbox{Ext}\,^i_R(\mbox{Tr}\,K_{\eta ^2},N)=0\) 
for all \(i>0\) .
By [1]}, from the exact sequence \(0\rightarrow M\rightarrow K_{\eta ^2}\rightarrow \Omega ^{2q+1}_R(M)\rightarrow 0\) , we obtain the following exact sequence
\(0\rightarrow {(\Omega ^{2q+1}_R(M))}^*\rightarrow {(K_{\eta ^2})}^*\rightarrow M^*\rightarrow \Omega ^{2q+1}_R(M))\rightarrow K_{\eta ^2})\rightarrow M)\rightarrow 0,\) 
where \(\underline{M)}\cong \underline{\mbox{Tr}\,M}\) , \(\underline{\Omega ^{2q+1}_R(M))}\cong \underline{\mathcal {T}_{2q+2}M}\)  and \(\underline{K_{\eta ^2})}\cong \underline{\mbox{Tr}\,K_{\eta ^2}}\) . As \(\mbox{Ext}\,^{2q+2}_R(M,R)=0\) , we get the exact sequence
\(0\rightarrow \Omega ^{2q+1}_R(M))\rightarrow K_{\eta ^2})\rightarrow M)\rightarrow 0.\) 
The exact sequence (REF ), induces a long exact sequence
\(\cdots \rightarrow \mbox{Ext}\,^i_R(M),N)\rightarrow \mbox{Ext}\,^i_R(K_{\eta ^2}),N)\rightarrow \mbox{Ext}\,^i_R(\Omega ^{2q+1}_R(M)),N)\rightarrow \cdots \) 
of cohomology modules.
As \(\mbox{Ext}\,^i_R(\mbox{Tr}\,K_{\eta ^2},N)=0\)  and \(\underline{\mathcal {T}_iM}\cong \underline{\Omega \mathcal {T}_{i+1}M}\)  for all \(i>0\) , by (REF ) we get the following isomorphisms.
\(\mbox{Ext}\,^i_R(\mathcal {T}_{2q+2}M,N)\cong \mbox{Ext}\,^{i+1}_R(\mbox{Tr}\,M,N)\cong \mbox{Ext}\,^{2q+i+2}_R(\mathcal {T}_{2q+2}M,N) \text{ for all } i>0.\) 
If \(q=0\)  then by (REF ), (REF ) and (REF ), it is obvious that \(\mbox{Ext}\,^i_R(\mbox{Tr}\,M,N)=0\)  for all \(i>0\) . Now if \(q>0\)  then
by (REF ) and (REF ),
\(\mbox{Ext}\,^{2q+2}_R(\mathcal {T}_{q+1}M,N)\cong \mbox{Ext}\,^{3q+3}_R(\mathcal {T}_{2q+2}M,N)\cong \mbox{Ext}\,^{q+1}_R(\mathcal {T}_{2q+2}M,N)=0\) 
Therefore by (REF ) and (REF ), \(\mbox{Ext}\,^i_R(\mathcal {T}_{q+1}M,N)=0\)  for all \(i>0\)  and so \(\mbox{Ext}\,^i_R(\mbox{Tr}\,M,N)=0\)  for all \(i>0\) .
